+1 Even more readable! I posted my question+answer just thinking it would be helpful for my fellow Linux users, but now I am getting a lot of benefit from the other answers :-)
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Nicolas RaoulAug 12 '14 at 6:52

3

There's no reason to be using let. It's not any more standard or portable than (( a = 3 * (2 + 1) )) (both come from ksh and are only available in ksh, bash and zsh) and it's less legible or easy to quote. Use a=$((3 * (2 + 1))) to be portable.
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Stéphane ChazelasAug 12 '14 at 15:27

2

I'm not saying it's wrong, I'm just saying it should not be used as there are better alternatives (one for legibility ((a = 3 * (2 + 1) )), one for portability a=$((3 * (2 + 1)))), so it's not a note against you or your answer but against it being the selected answer and top-scorer.
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Stéphane ChazelasAug 12 '14 at 15:48

What Nicolas illustrates but doesn’t explain is that the tokens on the expr command line must be separated by spaces; so; for example, expr 3 "*" "(2" "+" "1)"will not work. (Also, BTW, you probably don’t need to quote the +.)
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G-ManAug 12 '14 at 17:28

The parentheses aren't keywords like while and [[, they're syntax. If they were keywords, they wouldn't be interpreted as such in command arguments. You need quotes so that bash doesn't parse them but instead sees a string literal.
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GillesAug 13 '14 at 1:31

Arithmetic Expansion
Arithmetic expansion allows the evaluation of an arithmetic expression and the substitution of the result. The format for arithmetic expansion is:

$((expression))

The expression is treated as if it were within double quotes, but a double quote inside the parentheses is not treated specially. All tokens in the expression undergo parameter expansion, string expansion, command substitution, and quote removal. Arithmetic expansions may be nested.

The evaluation is performed according to the rules listed below under ARITHMETIC EVALUATION. If expression is invalid, bash prints a message indicating failure and no substitution occurs.

POSIX defines the $((...)) expansion operator. So you can use that in all POSIX compliant shells (the sh of all modern Unix-likes, dash, bash, yash, mksh, zsh, posh, ksh...).

a=$(( 3 * (2 + 1) ))
a=$((3*(2+1)))

ksh also introduced a let builtin which is passed the same kind of arithmetic expression, doesn't expand into something but returns an exit status based on whether the expression resolves to 0 or not, like in expr:

if let 'a = 3 * (2 + 1)'; then
echo "$a is non-zero"
fi

However, as the quoting makes it awkward and not very legible (not to the same extent as expr of course), ksh also introduced a ((...)) alternative form:

let and ((...)) are only available in ksh, zsh and bash. The $((...)) syntax should be preferred if portability to other shells is needed, expr is only needed for pre-POSIX Bourne-like shells (typically the Bourne shell or early versions of the Almquist shell).

On the non-Bourne front, there are a few shells with built-in arithmetic operator:

csh/tcsh (actually the first Unix shell with arithmetic evaluation built-in):

@ a = 3 * (2 + 1)

akanga (based on rc)

a = $:'3 * (2 + 1)'

as a history note, the original version of the Almquist shell, as posted on usenet in 1989 had an expr builtin (actually merged with test), but it was removed later.

expr is an external command, it is not special shell syntax. Therefore, if you want expr to see shell special characters, you need to protect them from shell parsing by quoting them. Furthermore, expr needs each number and operator to be passed as a separate parameter. Thus:

expr 3 \* \( 2 + 1 \)

Unless you're working on an antique unix system from the 1970s or 1980s, there is very little reason to use expr. In the old days, shells didn't have a built-in way to perform arithmetic, and you had to call the expr utility instead. All POSIX shells have built-in arithmetic via the arithmetic expansion syntax.

echo $((3 * (2 + 1)))

The construct $((…)) expands to the result of the arithmetic expression (written in decimal). Bash, like most shells, supports only integer arithmetic modulo 264 (or modulo 232 for older versions of bash and some other shells on 32-bit machines).

Bash offers an additional convenience syntax when you want to perform assignments or to test whether an expression is 0 but don't care about the result. This construct also exists in ksh and zsh but not in plain sh.

((x = 3 * (2+1)))
echo "$x"
if ((x > 3)); then …

In addition to integer arithmetic, expr offers a few string manipulation functions. These too are subsumed by features of POSIX shells, except for one: expr STRING : REGEXP tests whether the string matches the specified regexp. A POSIX shell cannot do this without external tools, but bash can with [[ STRING =~ REGEXP ]] (with a different regexp syntax — expr is a classic tool and uses BRE, bash uses ERE).

Unless you're maintaing scripts that run on 20-year-old systems, you don't need to know that expr ever existed. Use shell arithmetic.

expr foo : '\(.\)' also does text extraction. bash's BASH_REMATCH achieves something similar. It also does string comparison, which POSIX [ does not do (though one could imagine ways to use sort for that).
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Stéphane ChazelasAug 15 '14 at 11:25